where \(\mathbf {T} \in \mathbb {R}^{n \times n}\)  is the generalized transition matrix and \(\theta _{k}\)  is the weighting coefficient which determines the ratio of global-local information. To guarantee convergence, some stricter conditions are considered which requires that \(\sum _{k=0}^{\infty } \theta _{k}=1, \theta _{k} \in [0,1]\)  and the eigenvalues of \(\mathbf {T}\)  are bounded by \(\lambda _i \in [0,1]\) . By applying specific definitions of \(\mathbf {T}\)  and \(\theta \) , different instantiations of graph diffusion can be computed. For instance, Personalized PageRank (PPR) [1]} and the heat kernel [2]} are two popular examples of diffusion. Concretely, PPR chooses \(\mathbf {T}=\mathbf {A} \mathbf {D}^{-1}\)  and \(\theta _{k}=\alpha (1-\alpha )^{k}\) , where \(\mathbf {D} \in \mathbb {R}^{n \times n}\)  is the diagonal degree matrix and \(\alpha \in (0,1)\)  is the teleport probability. The heat kernel considers \(\mathbf {T}=\mathbf {A} \mathbf {D}^{-1}\)  and \(\theta _{k}=e^{-t} t^{k} / k !\) , where \(\beta \)  is the diffusion time. To avoid multiple steps of iteration, the closed-form solutions to PPR and heat kernel are formulated as Eq. REF  and Eq. REF  respectively.
\(\mathbf {S}^{\mathrm {PPR}}=\alpha \left(\mathbf {I}_{n}-(1-\alpha ) \mathbf {D}^{-1 / 2} \mathbf {A} \mathbf {D}^{-1 / 2}\right)^{-1}\) 
\(\mathbf {S}^{\text{heat }}=\exp \left(\beta \mathbf {A} \mathbf {D}^{-1}- \beta \right)\) 
